Group #13102924 — subgroups

Subgroups — #13102924

Order 1600.

608 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11717517	C₂	1	1	yes	maximal, characteristic
2	C₅²×#124	C₂	1	1	yes	maximal, characteristic, complemented
3	#11717262	C₂	1	2	yes	maximal, complemented
4	#11716700	C₂	1	1	yes	maximal, characteristic, complemented
5	#11716672	C₂	1	2	yes	maximal, complemented
6	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic, complemented
7	C₂×C₅²×D₄	C₂²	1	1	yes	characteristic
8	C₂×C₂²⋊(C₅⋊D₅)	—	2	2	no	—
9	#87902	—	2	2	no	—
10	D₄×C₅⋊D₅	—	2	2	no	—
11	#87900	C₂²	1	2	yes	—
12	C₅²×C₂²⋊C₄	—	2	2	no	—
13	C₄²×C₅²	C₂²	1	1	yes	characteristic, complemented
14	#87812	—	2	2	no	—
15	#87811	—	2	2	no	—
16	#87804	C₂²	1	2	yes	complemented
17	#65350	—	5	6	no	maximal
18	C₂³×C₅²	D₄	1	2	yes	—
19	C₂²×C₅⋊D₅	—	2	2	no	—
20	C₅²×Q₈	—	2	2	no	—
21	C₅²×D₄	—	2	2	no	—
22	C₂×C₄×C₅²	—	2	1	no	—
23	C₂×C₄×C₅²	—	2	1	no	—
24	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
25	C₂²⋊(C₅⋊D₅)	—	2	4	no	—
26	C₂²⋊(C₅⋊D₅)	—	2	2	no	—
27	C₂×#1039	—	2	2	no	—
28	C₄⋊(C₅⋊D₅)	D₄	1	2	yes	—
29	C₄×C₅⋊D₅	—	2	2	no	—
30	#6045	D₄	1	2	yes	—
31	#6029	—	2	2	no	—
32	#4144	—	5	6	no	—
33	C₅×#124	D₅	1	6	yes	complemented
34	#4088	—	5	12	no	—
35	#3965	—	5	6	no	—
36	#3937	—	5	12	no	—
37	C₂²×C₅²	—	2	2	no	—
38	C₂²×C₅²	—	4	2	no	—
39	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
40	C₂×C₅⋊D₅	—	2	2	no	—
41	C₂×C₅⋊D₅	—	8	2	no	—
42	C₄×C₅²	—	4	1	no	—
43	C₄×C₅²	—	4	1	no	—
44	C₄×C₅²	C₂×D₄	1	2	yes	—
45	#1039	—	2	2	no	—
46	C₂×C₅×Q₈	C₂×D₅	1	6	yes	complemented
47	C₂×C₅×D₄	C₂×D₅	1	6	yes	—
48	C₂×C₂²⋊D₅	—	10	12	no	—
49	#711	—	10	12	no	—
50	D₄×D₅	—	10	12	no	—
51	#709	—	5	12	no	—
52	C₅×C₂²⋊C₄	—	2	12	no	—
53	C₄²×C₅	C₂×D₅	1	6	yes	complemented
54	#689	—	10	12	no	—
55	#688	—	10	12	no	—
56	#681	—	5	12	no	—
57	#455	—	25	1	no	Sylow
58	C₂×C₅²	—	2	1	no	—
59	C₂×C₅²	—	4	2	no	—
60	C₂×C₅²	#120	1	1	yes	characteristic
61	C₅⋊D₅	—	4	2	no	—
62	C₂³×C₅	C₂²⋊D₅	1	12	yes	—
63	C₂²×D₅	—	10	12	no	—
64	C₅×Q₈	—	2	12	no	—
65	C₅×D₄	—	2	12	no	—
66	C₂×C₄×C₅	—	2	6	no	—
67	C₂×C₄×C₅	—	2	6	no	—
68	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
69	C₂²⋊D₅	—	10	24	no	—
70	C₂²⋊D₅	—	10	12	no	—
71	C₂×Dic₅	—	10	12	no	—
72	D₂₀	—	5	12	no	—
73	C₄×D₅	—	10	12	no	—
74	Dic₁₀	—	5	12	no	—
75	#168	—	10	12	no	—
76	H(5,𝔽₂)	—	25	1	no	—
77	#137	—	25	2	no	—
78	#124	C₅⋊D₅	1	1	yes	characteristic, complemented
79	#104	—	25	2	no	—
80	#100	—	25	1	no	—
81	C₅²	#455	1	1	yes	characteristic, complemented, Sylow
82	C₂²×C₅	—	2	12	no	—
83	C₂²×C₅	—	4	12	no	—
84	C₂²×C₅	C₂×C₂²⋊D₅	1	6	yes	—
85	C₂×D₅	—	10	12	no	—
86	C₂×D₅	—	40	12	no	—
87	C₄×C₅	—	4	6	no	—
88	C₄×C₅	—	4	6	no	—
89	C₄×C₅	D₄×D₅	1	12	yes	—
90	Dic₅	—	10	12	no	—
91	#41	—	25	2	no	—
92	#41	—	50	2	no	—
93	C₂×Q₈	C₂×C₅⋊D₅	1	1	yes	characteristic, complemented
94	C₂×D₄	—	50	2	no	—
95	C₂×D₄	—	50	2	no	—
96	C₂×D₄	C₂×C₅⋊D₅	1	1	yes	characteristic
97	Q₁₆	—	50	2	no	—
98	SD₁₆	—	50	2	no	—
99	M₁₆	—	25	2	no	—
100	C₂²⋊C₄	—	2	2	no	—
101	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic, complemented
102	C₂×C₅	—	2	6	no	—
103	C₂×C₅	—	4	12	no	—
104	C₂×C₅	#4083	1	6	yes	—
105	D₅	—	20	12	no	—
106	C₂³	—	50	2	no	—
107	C₂³	C₂²⋊(C₅⋊D₅)	1	2	yes	—
108	Q₈	—	2	2	no	—
109	Q₈	—	25	2	no	—
110	D₄	—	2	2	no	—
111	D₄	—	25	2	no	—
112	D₄	—	50	4	no	—
113	D₄	—	50	2	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	50	2	no	—
117	C₂×C₄	—	50	2	no	—
118	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
119	C₈	—	50	2	no	—
120	C₅	#65350	1	6	yes	complemented
121	C₂²	—	2	2	no	—
122	C₂²	—	4	2	no	—
123	C₂²	—	50	2	no	—
124	C₂²	—	200	2	no	—
125	C₂²	C₂×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
126	C₄	—	4	1	no	—
127	C₄	—	4	1	no	—
128	C₄	—	50	2	no	—
129	C₄	D₄×C₅⋊D₅	1	2	yes	—
130	C₂	—	2	1	no	—
131	C₂	—	4	2	no	—
132	C₂	—	100	2	no	—
133	C₂	#11717257	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11717517

C₂

yes

maximal, characteristic

C₅²×#124

C₂

yes

maximal, characteristic, complemented

#11717262

C₂

yes

maximal, complemented

#11716700

C₂

yes

maximal, characteristic, complemented

#11716672

C₂

yes

maximal, complemented

C₂×C₅²×Q₈

C₂²

yes

characteristic, complemented

C₂×C₅²×D₄

C₂²

yes

characteristic

C₂×C₂²⋊(C₅⋊D₅)

—

#87902

—

D₄×C₅⋊D₅

—

#87900

C₂²

yes

—

C₅²×C₂²⋊C₄

—

C₄²×C₅²

C₂²

yes

characteristic, complemented

#87812

—

#87811

—

#87804

C₂²

yes

complemented

#65350

—

maximal

C₂³×C₅²

D₄

yes

—

C₂²×C₅⋊D₅

—

C₅²×Q₈

—

C₅²×D₄

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂²⋊(C₅⋊D₅)

—

C₂²⋊(C₅⋊D₅)

—

C₂×#1039

—

C₄⋊(C₅⋊D₅)

D₄

yes

—

C₄×C₅⋊D₅

—

#6045

D₄

yes

—

#6029

—

#4144

—

C₅×#124

D₅

yes

—

—

—

C₂²×C₅²

—

C₂²×C₅²

—

C₂²×C₅²

C₂×D₄

yes

characteristic

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₄×C₅²

—

C₄×C₅²

—

C₄×C₅²

C₂×D₄

yes

—

#1039

—

C₂×C₅×Q₈

C₂×D₅

yes

complemented

C₂×C₅×D₄

C₂×D₅

yes

—

C₂×C₂²⋊D₅

—

#711

—

D₄×D₅

—

#709

—

C₅×C₂²⋊C₄

—

C₄²×C₅

C₂×D₅

yes

—

—

—

—

C₂×C₅²

—

C₂×C₅²

—

C₂×C₅²

#120

yes

characteristic

C₅⋊D₅

—

C₂³×C₅

C₂²⋊D₅

yes

—

C₂²×D₅

—

C₅×Q₈

—

C₅×D₄

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂²⋊D₅

—

C₂²⋊D₅

—

C₂×Dic₅

—

D₂₀

—

C₄×D₅

—

Dic₁₀

—

—

—

—

C₅⋊D₅

yes

characteristic, complemented

#104

—

#100

—

C₅²

#455

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₂×C₂²⋊D₅

yes

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

D₄×D₅

yes

—

Dic₅

—

#41

—

#41

—

C₂×Q₈

C₂×C₅⋊D₅

yes

characteristic, complemented

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×C₅⋊D₅

yes

characteristic

Q₁₆

—

SD₁₆

—

M₁₆

—

100

C₂²⋊C₄

—

101

C₄²

C₂×C₅⋊D₅

yes

characteristic, complemented

102

C₂×C₅

—

103

C₂×C₅

—

104

C₂×C₅

#4083

yes

—

105

D₅

—

106

C₂³

—

107

C₂³

C₂²⋊(C₅⋊D₅)

yes

—

108

Q₈

—

109

Q₈

—

110

D₄

—

111

D₄

—

112

D₄

—

113

D₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

C₂²×C₅⋊D₅

yes

characteristic

119

C₈

—

120

C₅

#65350

yes

complemented

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

200

—

125

C₂²

C₂×C₂²⋊(C₅⋊D₅)

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

D₄×C₅⋊D₅

yes

—

130

C₂

—

131

C₂

—

132

C₂

—

100

—

133

C₂

#11717257

yes

characteristic

	subgroup	G/H
center	C₂	#11717257
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂²×C₅⋊D₅
Fitting	C₅²×#124	C₂
socle	C₂×C₅²	#120
prime radical	#11717517	C₂
hypercenter	#124	C₅⋊D₅
hypocenter	C₅²	#455
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #124

subgroup

G/H

center

C₂

#11717257

commutator, cosocle

C₂×C₄×C₅²

C₂³

Frattini

C₂×C₄

C₂²×C₅⋊D₅

Fitting

C₅²×#124

C₂

C₂×C₅²

C₂

C₅⋊D₅

C₅²

G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #124

N	H ≅ G/N	type
2. C₅²×#124	C₂	semidirect
3. #11717262	C₂	semidirect
4. #11716700	C₂	semidirect
5. #11716672	C₂	semidirect
6. C₂×C₅²×Q₈	C₂²	semidirect
13. C₄²×C₅²	C₂²	semidirect
16. #87804	C₂²	semidirect
33. C₅×#124	D₅	semidirect
46. C₂×C₅×Q₈	C₂×D₅	semidirect
53. C₄²×C₅	C₂×D₅	semidirect
78. #124	C₅⋊D₅	semidirect
81. C₅²	#455	semidirect
93. C₂×Q₈	C₂×C₅⋊D₅	semidirect
101. C₄²	C₂×C₅⋊D₅	semidirect
120. C₅	#65350	semidirect

H ≅ G/N

type

2. C₅²×#124

C₂

semidirect

3. #11717262

C₂

semidirect

4. #11716700

C₂

semidirect

5. #11716672

C₂

semidirect

6. C₂×C₅²×Q₈

C₂²

semidirect

13. C₄²×C₅²

C₂²

semidirect

16. #87804

C₂²

semidirect

33. C₅×#124

D₅

semidirect

46. C₂×C₅×Q₈

C₂×D₅

semidirect

53. C₄²×C₅

C₂×D₅

semidirect

78. #124

C₅⋊D₅

semidirect

81. C₅²

#455

semidirect

93. C₂×Q₈

C₂×C₅⋊D₅

semidirect

101. C₄²

C₂×C₅⋊D₅

semidirect

120. C₅

#65350

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#11717517	2
C₅²×#124	2
#11717262	2
#11716700	2
#11716672	2
#65350	5

subgroup

index

C₅²×#124

Subgroups — #13102924

Conjugacy classes of subgroups

Complements

Maximal subgroups